Infinite Games · a multiple reduction copy machine
Copy!
Put a picture on the glass. The machine prints shrunken copies of it, then copies the copy, again and again, until the picture is gone and a fractal is left.
▶ Start copyingSame blueprint, any picture, same result
Pick a source and a blueprint, then step through the passes. Each pass replaces every copy on the page with several smaller copies of the whole page. By pass 6 or so, you can't tell which picture you started with.
A photocopier with several lenses
The multiple reduction copy machine is a thought experiment from Heinz-Otto Peitgen, Hartmut Jürgens and Dietmar Saupe, used in Chaos and Fractals (1992) to explain iterated function systems. Picture a copier fitted with several lenses instead of one. Each lens shrinks the original and lays it somewhere on the page. Feed the page back in and let it run.
Each lens is an affine contraction: it scales by less than 100%, and may also rotate, skew, flip and shift. In the game, each lens is one frame on the blueprint.
Why the source stops mattering
Every lens shrinks, so any two starting pictures get closer with each pass. John Hutchinson proved in 1981 that the process settles on exactly one shape, the attractor, whatever you start with. All that survives of your picture is its color at the finest scale.
Running it backwards
Michael Barnsley's collage theorem turns this around: cover a shape with shrunken copies of itself, and those copies form a blueprint that regrows the shape from anything. The fern is four lenses, one squashed almost flat to make the stem.
Fractional dimension
Sierpiński's triangle (1915) is three lenses at 50%. Each pass triples the copies and halves their size. Its dimension D is the number for which three half-size copies rebuild the whole.
The game's dimension readout solves the same equation, Σ rᵢᴰ = 1, for any blueprint you build, where rᵢ is each copy's reduction. A line has dimension 1 and a filled square 2; the triangle sits between them.
The canvas is the instrument
- Touch
- Drag, pinch, twist, double-tap and press the canvas. Each gesture bends the blueprint a different way. There are eight to discover, and the machine names each one as you find it.
- Play
- The round button makes the blueprint drift, so the fractal slowly morphs.
- Source
- 91 symbols to copy (signs, nature, objects, faces, shapes), or upload your own image. Transparent backgrounds work best.
- Blueprint
- Place each copy exactly: reduce, rotate, skew, flip, move. Presets, undo, and saved compositions.
- Adjust
- Three looks (Living Canvas, Optical Lab, Playground), iteration count, and two print styles: Copies draws the picture at every leaf; Dust plots the attractor point by point.
- Explore
- Dive into the attractor, pan and zoom, record a 10-second clip, save an image, or share a link to your blueprint.
- Keys
- Space play · D dive · R record · S source · B blueprint · F fit · Ctrl/⌘-Z undo
Works on phone or desktop. Opens on the Sierpiński triangle.